Claudia Castro-Castro
Math 283 Spring 2020
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DEFINITION Given a parameterization of the surface \[ \mathbf{\vec{r}}(u, v) = x(u,v)\mathbf{\hat{i}}+y(u,v)\mathbf{\hat{j}}+z(u,v)\mathbf{\hat{k}} \] the parameter domain is the set of points in the \( uv \)-plane that can be substituted into \( \mathbf{\vec{r}} \)
\[ \mathbf{\vec{r}}(u, v)= \langle 2 \cos u,2 \sin u, v\rangle\\ D=[0,2\pi]\times[-3,3] \]
\[ \mathbf{\vec{r}}(u, v)= \langle u, v, v^2\rangle\\ D=[-2,2]\times[-2,2] \]
Surface area \[ A(S)=\iint\limits_D ||\mathbf{\vec{r}}_u \times \mathbf{\vec{r}}_v||dA \] \( D \) is the parameter domain